Pub Date : 2024-06-01DOI: 10.4208/cicp.oa-2023-0078
Lizuo Liu,Kamaljyoti Nath, Wei Cai
In this paper, we propose a DeepONet structure with causality to represent causal linear operators between Banach spaces of time-dependent signals. The theorem of universal approximations to nonlinear operators proposed in [5] is extended to operators with causalities, and the proposed Causality-DeepONet implements the physical causality in its framework. The proposed Causality-DeepONet considers causality (the state of the system at the current time is not affected by that of the future, but only by its current state and past history) and uses a convolution-type weight in its design. To demonstrate its effectiveness in handling the causal response of a physical system, the Causality-DeepONet is applied to learn the operator representing the response of a building due to earthquake ground accelerations. Extensive numerical tests and comparisons with some existing variants of DeepONet are carried out, and the Causality-DeepONet clearly shows its unique capability to learn the retarded dynamic responses of the seismic response operator with good accuracy.
{"title":"A Causality-DeepONet for Causal Responses of Linear Dynamical Systems","authors":"Lizuo Liu,Kamaljyoti Nath, Wei Cai","doi":"10.4208/cicp.oa-2023-0078","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0078","url":null,"abstract":"In this paper, we propose a DeepONet structure with causality to represent\u0000causal linear operators between Banach spaces of time-dependent signals. The theorem of universal approximations to nonlinear operators proposed in [5] is extended\u0000to operators with causalities, and the proposed Causality-DeepONet implements the\u0000physical causality in its framework. The proposed Causality-DeepONet considers\u0000causality (the state of the system at the current time is not affected by that of the future, but only by its current state and past history) and uses a convolution-type weight\u0000in its design. To demonstrate its effectiveness in handling the causal response of a\u0000physical system, the Causality-DeepONet is applied to learn the operator representing\u0000the response of a building due to earthquake ground accelerations. Extensive numerical tests and comparisons with some existing variants of DeepONet are carried out,\u0000and the Causality-DeepONet clearly shows its unique capability to learn the retarded\u0000dynamic responses of the seismic response operator with good accuracy.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"45 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141502901","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents a novel 3-D full electromagnetic particle-in-cell (PIC) code called JefiPIC, which uses Jefimenko’s equations as the electromagnetic (EM) field solver through a full-space integration method. Leveraging the power of state-of-the-art graphic processing units (GPUs), we have made the challenging integral task of PIC simulations achievable. Our proposed code offers several advantages by utilizing the integral method. Firstly, it offers a natural solution for modeling non-neutral plasmas without the need for pre-processing such as solving Poisson’s equation. Secondly, it eliminates the requirement for designing elaborate boundary layers to absorb fields and particles. Thirdly, it maintains the stability of the plasma simulation regardless of the time step chosen. Lastly, it does not require strict charge-conservation particle-to-grid apportionment techniques or electric field divergence amendment algorithms, which are commonly used in finite-difference time-domain (FDTD)-based PIC simulations. To validate the accuracy and advantages of our code, we compared the evolutions of particles and fields in different plasma systems simulated by three other codes. Our results demonstrate that the combination of Jefimenko’s equations and the PIC method can produce accurate particle distributions and EM fields in open-boundary plasma systems. Additionally, our code is able to accomplish these computations within an acceptable execution time. This study highlights the effectiveness and efficiency of JefiPIC, showing its potential for advancing plasma simulations.
{"title":"JefiPIC: A 3-D Full Electromagnetic Particle-in-Cell Simulator Based on Jefimenko’s Equations on GPU","authors":"Jian-Nan Chen,Jun-Jie Zhang,Hai-Liang Qiao,Xue-Ming Li, Yong-Tao Zhao","doi":"10.4208/cicp.oa-2023-0156","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0156","url":null,"abstract":"This paper presents a novel 3-D full electromagnetic particle-in-cell (PIC)\u0000code called JefiPIC, which uses Jefimenko’s equations as the electromagnetic (EM) field\u0000solver through a full-space integration method. Leveraging the power of state-of-the-art graphic processing units (GPUs), we have made the challenging integral task of\u0000PIC simulations achievable. Our proposed code offers several advantages by utilizing the integral method. Firstly, it offers a natural solution for modeling non-neutral\u0000plasmas without the need for pre-processing such as solving Poisson’s equation. Secondly, it eliminates the requirement for designing elaborate boundary layers to absorb\u0000fields and particles. Thirdly, it maintains the stability of the plasma simulation regardless of the time step chosen. Lastly, it does not require strict charge-conservation\u0000particle-to-grid apportionment techniques or electric field divergence amendment algorithms, which are commonly used in finite-difference time-domain (FDTD)-based\u0000PIC simulations. To validate the accuracy and advantages of our code, we compared\u0000the evolutions of particles and fields in different plasma systems simulated by three\u0000other codes. Our results demonstrate that the combination of Jefimenko’s equations\u0000and the PIC method can produce accurate particle distributions and EM fields in open-boundary plasma systems. Additionally, our code is able to accomplish these computations within an acceptable execution time. This study highlights the effectiveness\u0000and efficiency of JefiPIC, showing its potential for advancing plasma simulations.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"80 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a model-data asymptotic-preserving neural network(MD-APNN) method to solve the nonlinear gray radiative transfer equations (GRTEs). The system is challenging to be simulated with both the traditional numerical schemes and the vanilla physics-informed neural networks (PINNs) due to the multiscale characteristics. Under the framework of PINNs, we employ a micro-macro decomposition technique to construct a new asymptotic-preserving (AP) loss function, which includes the residual of the governing equations in the micro-macro coupled form, the initial and boundary conditions with additional diffusion limit information, the conservation laws, and a few labeled data. A convergence analysis is performed for the proposed method, and a number of numerical examples are presented to illustrate the efficiency of MD-APNNs, and particularly, the importance of the AP property in the neural networks for the diffusion dominating problems. The numerical results indicate that MD-APNNs lead to a better performance than APNNs or pure Data-driven networks in the simulation of the nonlinear non-stationary GRTEs.
{"title":"A Model-Data Asymptotic-Preserving Neural Network Method Based on Micro-Macro Decomposition for Gray Radiative Transfer Equations","authors":"Hongyan Li,Song Jiang,Wenjun Sun,Liwei Xu, Guanyu Zhou","doi":"10.4208/cicp.oa-2022-0315","DOIUrl":"https://doi.org/10.4208/cicp.oa-2022-0315","url":null,"abstract":"We propose a model-data asymptotic-preserving neural network(MD-APNN) method to solve the nonlinear gray radiative transfer equations (GRTEs). The\u0000system is challenging to be simulated with both the traditional numerical schemes\u0000and the vanilla physics-informed neural networks (PINNs) due to the multiscale characteristics. Under the framework of PINNs, we employ a micro-macro decomposition technique to construct a new asymptotic-preserving (AP) loss function, which includes the residual of the governing equations in the micro-macro coupled form, the\u0000initial and boundary conditions with additional diffusion limit information, the conservation laws, and a few labeled data. A convergence analysis is performed for the\u0000proposed method, and a number of numerical examples are presented to illustrate the\u0000efficiency of MD-APNNs, and particularly, the importance of the AP property in the\u0000neural networks for the diffusion dominating problems. The numerical results indicate that MD-APNNs lead to a better performance than APNNs or pure Data-driven\u0000networks in the simulation of the nonlinear non-stationary GRTEs.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"5 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141502900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-01DOI: 10.4208/cicp.oa-2023-0203
Ting Wang,Yuzhi Zhou, Aihui Zhou
In this paper, we apply the effective potentials in the localization landscape theory (Filoche et al., 2012, Arnold et al., 2016) to study the spectral properties of the incommensurate systems. We uniquely develop a plane wave framework for the effective potentials of the incommensurate systems. And utilizing the effective potentials represented by the plane wave, the location of the electron density can be inferred. Moreover, the spectral distribution can be obtained from the effective potential version of Weyl’s law. We perform some numerical experiments on some typical incommensurate systems, showing that the effective potential provides an alternative tool for investigating the localization and spectral properties of the incommensurate systems, without solving the eigenvalue problem explicitly.
{"title":"Localization in the Incommensurate Systems: A Plane Wave Study via Effective Potentials","authors":"Ting Wang,Yuzhi Zhou, Aihui Zhou","doi":"10.4208/cicp.oa-2023-0203","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0203","url":null,"abstract":"In this paper, we apply the effective potentials in the localization landscape\u0000theory (Filoche et al., 2012, Arnold et al., 2016) to study the spectral properties of the\u0000incommensurate systems. We uniquely develop a plane wave framework for the effective potentials of the incommensurate systems. And utilizing the effective potentials\u0000represented by the plane wave, the location of the electron density can be inferred.\u0000Moreover, the spectral distribution can be obtained from the effective potential version\u0000of Weyl’s law. We perform some numerical experiments on some typical incommensurate systems, showing that the effective potential provides an alternative tool for\u0000investigating the localization and spectral properties of the incommensurate systems,\u0000without solving the eigenvalue problem explicitly.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"29 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140940520","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-01DOI: 10.4208/cicp.oa-2023-0289
Tian Liang, Lin Fu
The development of high-order shock-capturing schemes is critical for compressible fluid simulations, in particular for cases where both shock waves and small-scale turbulence structures present. As one of the state-of-the-art high-order numerical schemes, the family of high-order targeted ENO (TENO) schemes proposed by Fu et al. [Journal of Computational Physics 305 (2016): 333-359] has been demonstrated to perform well for compressible gas dynamics on structured meshes and recently extended to unstructured meshes by Ji et al. [Journal of Scientific Computing 92(2022): 1-39]. In this paper, with the observation that the TENO scheme not only provides the high-order reconstructed data at the cell interface but also features the potential to separate the local flow scales in the wavenumber space, we propose a low-dissipation finite-volume TENO scheme with a new cell-interface flux evaluation strategy for unstructured meshes. The novelty originates from the fact that the local flow scales are classified, following a strong scale separation in the reconstruction process, as “very smooth” or not. When the corresponding large central-biased stencil for the targeted cell interface is judged to be “very smooth”, a low-dissipation Riemann solver, even the non-dissipative central flux scheme, is employed for the cell-interface flux computing. Otherwise, a dissipative approximate Riemann solver is employed to avoid spurious oscillations and achieve stable shock-capturing. Such a strategy provides separate control over the numerical dissipation of the high-order reconstruction process and the cell-interface flux calculation within a unified framework and leads to a resultant finite-volume method with extremely low-dissipation properties and good numerical robustness. Without parameter tuning case by case, a set of canonical benchmark simulations has been conducted to assess the performance of the proposed scheme.
{"title":"Finite-Volume TENO Scheme with a New Cell-Interface Flux Evaluation Strategy for Unstructured Meshes","authors":"Tian Liang, Lin Fu","doi":"10.4208/cicp.oa-2023-0289","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0289","url":null,"abstract":"The development of high-order shock-capturing schemes is critical for compressible fluid simulations, in particular for cases where both shock waves and small-scale turbulence structures present. As one of the state-of-the-art high-order numerical\u0000schemes, the family of high-order targeted ENO (TENO) schemes proposed by Fu et\u0000al. [Journal of Computational Physics 305 (2016): 333-359] has been demonstrated to\u0000perform well for compressible gas dynamics on structured meshes and recently extended to unstructured meshes by Ji et al. [Journal of Scientific Computing 92(2022):\u00001-39]. In this paper, with the observation that the TENO scheme not only provides\u0000the high-order reconstructed data at the cell interface but also features the potential to\u0000separate the local flow scales in the wavenumber space, we propose a low-dissipation\u0000finite-volume TENO scheme with a new cell-interface flux evaluation strategy for unstructured meshes. The novelty originates from the fact that the local flow scales are\u0000classified, following a strong scale separation in the reconstruction process, as “very\u0000smooth” or not. When the corresponding large central-biased stencil for the targeted\u0000cell interface is judged to be “very smooth”, a low-dissipation Riemann solver, even\u0000the non-dissipative central flux scheme, is employed for the cell-interface flux computing. Otherwise, a dissipative approximate Riemann solver is employed to avoid spurious oscillations and achieve stable shock-capturing. Such a strategy provides separate\u0000control over the numerical dissipation of the high-order reconstruction process and\u0000the cell-interface flux calculation within a unified framework and leads to a resultant\u0000finite-volume method with extremely low-dissipation properties and good numerical\u0000robustness. Without parameter tuning case by case, a set of canonical benchmark simulations has been conducted to assess the performance of the proposed scheme.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"191 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140940447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-01DOI: 10.4208/cicp.re-2023-0066
Qiang He,Yiqian Cheng,Fengming Hu,Weifeng Huang, Decai Li
In recent years, phase-field-based models for multiphase flows have gained significant popularity, particularly within the lattice Boltzmann (LB) community. These models typically use two lattice Boltzmann equations (LBEs), one for interface tracking and the other for solving hydrodynamic properties. However, for the purposes of this paper, we focus only on the LB model for hydrodynamics. Our goal is to undertake a comparative investigation into the differences between three classical hydrodynamic LB models proposed by Lee et al. [1], Liang et al. [2] and Fakhari et al. [3]. The interface-tracking equation used in this study is based on the conservative phase-field model. We provide a detailed derivation of the governing equations in each model using the Chapman-Enskog analysis. Additionally, three discretization methods for the interaction forces are introduced, and a modified method for the gradient term is proposed based on the nonequilibrium distribution method. The accuracy of three LB models in combination with four discretization methods is examined in this study. Based on the results, it appears that different combinations of models and methods are appropriate for different types of problems. However, some suggestions for the selection of hydrodynamic models and discrete methods for the gradient term are provided in this paper.
{"title":"A Comparative Study of Hydrodynamic Lattice Boltzmann Equation in Phase-Field-Based Multiphase Flow Models","authors":"Qiang He,Yiqian Cheng,Fengming Hu,Weifeng Huang, Decai Li","doi":"10.4208/cicp.re-2023-0066","DOIUrl":"https://doi.org/10.4208/cicp.re-2023-0066","url":null,"abstract":"In recent years, phase-field-based models for multiphase flows have gained\u0000significant popularity, particularly within the lattice Boltzmann (LB) community. These\u0000models typically use two lattice Boltzmann equations (LBEs), one for interface tracking\u0000and the other for solving hydrodynamic properties. However, for the purposes of this\u0000paper, we focus only on the LB model for hydrodynamics. Our goal is to undertake a\u0000comparative investigation into the differences between three classical hydrodynamic\u0000LB models proposed by Lee et al. [1], Liang et al. [2] and Fakhari et al. [3]. The interface-tracking equation used in this study is based on the conservative phase-field model.\u0000We provide a detailed derivation of the governing equations in each model using the\u0000Chapman-Enskog analysis. Additionally, three discretization methods for the interaction forces are introduced, and a modified method for the gradient term is proposed\u0000based on the nonequilibrium distribution method. The accuracy of three LB models\u0000in combination with four discretization methods is examined in this study. Based on\u0000the results, it appears that different combinations of models and methods are appropriate for different types of problems. However, some suggestions for the selection of\u0000hydrodynamic models and discrete methods for the gradient term are provided in this\u0000paper.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"3 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140886665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-01DOI: 10.4208/cicp.oa-2023-0309
Ping Zeng, Guanyu Zhou
We propose a linear decoupled positivity-preserving scheme for the chemotaxis-fluid system, which models the interaction between aerobic bacteria and the fluid flow surrounding them. This scheme comprises the finite element method (FEM) for the fluid equations on a regular triangulation and an upwind finite volume method (FVM) for the chemotaxis system on two types of dual mesh. The discrete cellular density and chemical concentration are represented as the piecewise constant functions on the dual mesh. They can also be equivalently expressed as the piecewise linear functions on the triangulation in the sense of mass-lumping. These discrete solutions are obtained by the upwind finite volume approximation satisfying the laws of positivity preservation and mass conservation. The finite element method is used to compute the numerical velocity in the triangulation, which is then used to determine the upwind-style numerical flux in the dual mesh. We analyze the $M$-property of the matrices from the discrete system and prove the well-posedness and the positivity-preserving property. By using the $L^p$-estimate of the discrete Laplace operators, semigroup analysis, and induction method, we are able to establish the optimal error estimates for chemical concentration, cellular density, and velocity field in $(l^∞(W^{1,p}), l^∞(L^p),l^∞(W^{1,p}))$-norm. Several numerical examples are presented to verify the theoretical results.
{"title":"The Positivity-Preserving Finite Volume Coupled with Finite Element Method for the Keller-Segel-Navier-Stokes Model","authors":"Ping Zeng, Guanyu Zhou","doi":"10.4208/cicp.oa-2023-0309","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0309","url":null,"abstract":"We propose a linear decoupled positivity-preserving scheme for the\u0000chemotaxis-fluid system, which models the interaction between aerobic bacteria and\u0000the fluid flow surrounding them. This scheme comprises the finite element method\u0000(FEM) for the fluid equations on a regular triangulation and an upwind finite volume\u0000method (FVM) for the chemotaxis system on two types of dual mesh. The discrete\u0000cellular density and chemical concentration are represented as the piecewise constant\u0000functions on the dual mesh. They can also be equivalently expressed as the piecewise linear functions on the triangulation in the sense of mass-lumping. These discrete solutions are obtained by the upwind finite volume approximation satisfying\u0000the laws of positivity preservation and mass conservation. The finite element method\u0000is used to compute the numerical velocity in the triangulation, which is then used\u0000to determine the upwind-style numerical flux in the dual mesh. We analyze the $M$-property of the matrices from the discrete system and prove the well-posedness and\u0000the positivity-preserving property. By using the $L^p$-estimate of the discrete Laplace\u0000operators, semigroup analysis, and induction method, we are able to establish the optimal error estimates for chemical concentration, cellular density, and velocity field in $(l^∞(W^{1,p}), l^∞(L^p),l^∞(W^{1,p}))$-norm. Several numerical examples are presented to verify the theoretical results.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"308 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140940615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-01DOI: 10.4208/cicp.oa-2024-0012
Philipp Krah,Xi-Yuan Yin,Julius Bergmann,Jean-Christophe Nave, Kai Schneider
We propose an efficient semi-Lagrangian characteristic mapping method for solving the one+one-dimensional Vlasov-Poisson equations with high precision on a coarse grid. The flow map is evolved numerically and exponential resolution in linear time is obtained. Global third-order convergence in space and time is shown and conservation properties are assessed. For benchmarking, we consider linear and nonlinear Landau damping and the two-stream instability. We compare the results with a Fourier pseudo-spectral method and results from the literature. The extreme fine-scale resolution features are illustrated showing the method’s capabilities to efficiently treat filamentation in fusion plasma simulations.
{"title":"A Characteristic Mapping Method for Vlasov-Poisson with Extreme Resolution Properties","authors":"Philipp Krah,Xi-Yuan Yin,Julius Bergmann,Jean-Christophe Nave, Kai Schneider","doi":"10.4208/cicp.oa-2024-0012","DOIUrl":"https://doi.org/10.4208/cicp.oa-2024-0012","url":null,"abstract":"We propose an efficient semi-Lagrangian characteristic mapping method for\u0000solving the one+one-dimensional Vlasov-Poisson equations with high precision on a\u0000coarse grid. The flow map is evolved numerically and exponential resolution in linear time is obtained. Global third-order convergence in space and time is shown and\u0000conservation properties are assessed. For benchmarking, we consider linear and nonlinear Landau damping and the two-stream instability. We compare the results with a\u0000Fourier pseudo-spectral method and results from the literature. The extreme fine-scale\u0000resolution features are illustrated showing the method’s capabilities to efficiently treat\u0000filamentation in fusion plasma simulations.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"41 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140940617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-01DOI: 10.4208/cicp.oa-2023-0249
Chunyu Chen,Long Chen,Xuehai Huang, Huayi Wei
This study investigates high-order face and edge elements in finite element methods, with a focus on their geometric attributes, indexing management, and practical application. The exposition begins by a geometric decomposition of Lagrange finite elements, setting the foundation for further analysis. The discussion then extends to $H$(div)-conforming and $H$(curl)-conforming finite element spaces, adopting variable frames across differing sub-simplices. The imposition of tangential or normal continuity is achieved through the strategic selection of corresponding bases. The paper concludes with a focus on efficient indexing management strategies for degrees of freedom, offering practical guidance to researchers and engineers. It serves as a comprehensive resource that bridges the gap between theory and practice.
{"title":"Geometric Decomposition and Efficient Implementation of High Order Face and Edge Elements","authors":"Chunyu Chen,Long Chen,Xuehai Huang, Huayi Wei","doi":"10.4208/cicp.oa-2023-0249","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0249","url":null,"abstract":"This study investigates high-order face and edge elements in finite element\u0000methods, with a focus on their geometric attributes, indexing management, and practical application. The exposition begins by a geometric decomposition of Lagrange\u0000finite elements, setting the foundation for further analysis. The discussion then extends to $H$(div)-conforming and $H$(curl)-conforming finite element spaces, adopting\u0000variable frames across differing sub-simplices. The imposition of tangential or normal\u0000continuity is achieved through the strategic selection of corresponding bases. The paper concludes with a focus on efficient indexing management strategies for degrees\u0000of freedom, offering practical guidance to researchers and engineers. It serves as a\u0000comprehensive resource that bridges the gap between theory and practice.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"29 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140940521","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-01DOI: 10.4208/cicp.oa-2023-0162
Zixuan Zhang,Yidao Dong,Yuanyang Zou,Hao Zhang, Xiaogang Deng
With continuous developments in various techniques, machine learning is becoming increasingly viable and promising in the field of fluid mechanics. In this article, we present a machine learning approach for enhancing the resolution and robustness of the weighted compact nonlinear scheme (WCNS). We employ a neural network as a weighting function in the WCNS scheme and follow a data-driven approach to train this neural network. Neural networks can learn a new smoothness measure and calculate a weight function inherently. To facilitate the machine learning task and train with fewer data, we integrate the prior knowledge into the learning process, such as a Galilean invariant input layer and CNS polynomials. The normalization in the Delta layer (the so-called Delta layer is used to calculate input features) ensures that the WCNS3-NN schemes achieve a scale-invariant property (Si-property) with an arbitrary scale of a function, and an essentially non-oscillatory approximation of a discontinuous function (ENO-property). The Si-property and ENO-property of the data-driven WCNS schemes are validated numerically. Several one- and two-dimensional benchmark examples, including strong shocks and shock-density wave interactions, are presented to demonstrate the advantages of the proposed method.
{"title":"A Data-Driven Scale-Invariant Weighted Compact Nonlinear Scheme for Hyperbolic Conservation Laws","authors":"Zixuan Zhang,Yidao Dong,Yuanyang Zou,Hao Zhang, Xiaogang Deng","doi":"10.4208/cicp.oa-2023-0162","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0162","url":null,"abstract":"With continuous developments in various techniques, machine learning is\u0000becoming increasingly viable and promising in the field of fluid mechanics. In this\u0000article, we present a machine learning approach for enhancing the resolution and robustness of the weighted compact nonlinear scheme (WCNS). We employ a neural\u0000network as a weighting function in the WCNS scheme and follow a data-driven approach to train this neural network. Neural networks can learn a new smoothness\u0000measure and calculate a weight function inherently. To facilitate the machine learning task and train with fewer data, we integrate the prior knowledge into the learning\u0000process, such as a Galilean invariant input layer and CNS polynomials. The normalization in the Delta layer (the so-called Delta layer is used to calculate input features)\u0000ensures that the WCNS3-NN schemes achieve a scale-invariant property (Si-property)\u0000with an arbitrary scale of a function, and an essentially non-oscillatory approximation of a discontinuous function (ENO-property). The Si-property and ENO-property\u0000of the data-driven WCNS schemes are validated numerically. Several one- and two-dimensional benchmark examples, including strong shocks and shock-density wave\u0000interactions, are presented to demonstrate the advantages of the proposed method.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"27 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941880","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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